The nucleus-nucleus potential

2.N

I

NuclearPhystcs A259 (1976) 157--172, (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprmt or microfilm without v,ratten permission from the pubhsher

THE NUCLEUS-NUCLEUS POTENTIAL ) H R. J A Q A M A N a n d A Z M E K J I A N

Department of Phystcs, Rutgers University, New Brunswtck, New Jersey 08903 Received 10 N o v e m b e r 1975 Abstract: U s i n g t h e generator coordinate m e t h o d to investigate the vahd~ty o f the use o f ,t potential to describe h e a v y ~on elastic scattering, it is f o u n d that, u n d e r certain conditions, art effective potentxal exists T h e s e conditions are generally satisfied by heavy ~ons, especmlly for the p o t e n t m l ' s tazl T h e p o t e n t m l o b t a i n e d includes a kinetic energy c o n t r i b u t i o n T h e p o t e n t m l ~s further investigated u s i n g a d e n s i t y - d e p e n d e n t delta interaction

1. Introduction The generator coordinate method ( G C M ) 1,2) offers one of the most practical frameworks for the description of nuclear reactions between heavy ~ons. There have been numerous investigations of thts method m recent years [see e g., refs. 3-6,8), and references c~ted in ref. 2)]. In this case one starts from the Griffin-Hill-Wheeler ( G H W ) equation and solves for wave functions m the continuum. These solutions have to be carried out numerically [see e.g refs. 3,4)] since the G H W equation is an integral equation; th~s prevents one f r o m gaming insight into the nature of the solution, especially since one is accustomed to solving a Schrbdlnger-type differential equation. One would also hke to find a connection between the solutions to the G H W equahon and the phenomenological optical potentials obtained by fitting the data m heavy ion scattering. Apart from these theoretical interests, the numerical lntegrat~on and solution of the G H W equation meets with some techmcal dlfficulhes 4). Furthermore, most calculations usmg the G C M have so far been hmited to very light ions (mostly e-s), and although probably not out of the question, ~ts apphcat~on to heavier ions would entail a huge amount of calculation that has been the mare obstacle so far. In this paper we will attempt to transform the G H W equation into a dlfferentml (Schr6dmger-type) equation that will allow us to get a better insight into the processes taking place and to obtain the solutions more easdy In particular, th~s method gives us an effectwe potentml for the interaction between two heavy ions that can be compared with phenomenolog~cal ophcal potentmls. In sect. 2 we review the G C M as it has been apphed to heavy ~on reachons, and then we use ~t in sect. 3 to get the differential equation for the wave funchon of relative motion and consequently an expression for the nucleus-nucleus potentml and the condition of ~ts vahdity Thts potentml IS then further invest:gated m sect 4, where t S u p p o r t e d m part by t h e N a t i o n a l Science F o u n d a t t o n (grant n o

157

N S F GP39126X)

158

H. R. JAQAMAN AND A Z MEKJIAN

the e-~ 60 potential is calculated using a density-dependent force, and then used to calculate cluster states m 2°Ne The ~-~60 example is chosen partly because it is easy to get analytic expressions for it and partly because of the avallablhty of other calculations In general, however, these calculations have to be done numerically, and the expressions given m subsect 4 2 pay special attenhon to this The 1 6 0 - 1 6 0 potentml is also discussed in subsect 4 2

2. Review of GC~VI for reactions In the G C M we start w~th a wave function of the form (.Y/is the antisymmetnzmg operator): A1 + A2

qb(x. ~) = ~¢ [ H

¢,(r,, ~t~)] = det

z=l

{¢,(r,,~)},

(1)

that has A~ + A 2 nucleons distributed into a number ). of potential wells (here two). each well characterized by a set =; of parameters The final many-body wave function ~s the hnear combination ~(x) = J- ~(x, a)f(~t)d~

(2)

Applying the Ritz varmtional principle to ~b. we get the generator coordinate ( G H W ) wave equation

f[H(~, ~')-El(a, ~t')]f(~t')d~' -

0.

(3a)

~')~

(3b)

where

II( ,

a')J =

{ 11 Uj

These involve mtegratlons over the A I +A2 smgle-partlcle coordinates {r,} m the system Although the use of the Ritz varlatlonal principle is acceptable for bound state solutlons, ~t does not necessarily follow that it can be used for continuum solutions, However, the use of the G H W equatlon can be justified for these soluhons w~th or wlthout the use of some varlattonal prmclple 4, s). It is also known that the G C M breaks down for reactions between two unequal fragments 4,6); therefore we w:ll limit ourselves to two equal fragments. In the case of two unequal fragments, the wave function has a center-of-mass ( c . m ) spunoslty m it (because the total c m motion does not drop out) but its effect would be smaller for heavy ions than for light ions, so that our results are expected to also apply falrly well to two unequal heavy ~ons The sets of parameters ~z characterize the positions of the wells plus any other propertles such as their deformation In the present work we are only interested m the posmons of the wells and so we need two parameters =h and ~z to describe these positions that would coinclde with the expectatlon values of the c.m of the two ions

NUCLEUS-NUCLEUS POTENTIAL

159

Since the total c.m m o t i o n drops out, we actually need one p a r a m e t e r • = ~ x - at2The G H W generating function for a system of two identical clusters as then

• (x;

me) =

~x(x~-ea) = d

m,,

A~ [ H ~ , ( r , - ~ ? ) ] = det {~b,(r,-ea)}, 2=

(1')

1,2.

The wave function (1') can be written

cb(x, . , , me) = q~ .i (R~ m )'x~CG(r--m)Oi"tO'z"t],

(4)

where G(r- ~t) as a f u n c t m n peaked at r = ~, and I ~ 1,,t , I[/2 ,,t are the internal wave funcUons for the two nuclei and depend only on internal coordinates. It can be easdy shown that 9(r) = J- dm G(r- ~)f(m),

(5)

m a y be regarded as the wave function of relative m o t i o n between the two ions by c o m p a r i n g (2) with the resonating group wave functaon using the f o r m (4) for q~. One of the first techn,cal dffficultaes m the G C M is that m a n y functions 9 ( 0 cannot be represented by eq (5) with a well-behaved f u n c t i o n f ( e ) ; f ( e ) does not always exist although 9(r) does 4). It would be desirable therefore to t r a n s f o r m the p r o b l e m into an equation governing 9(r) with no m e n t i o n off(m). All calculations done so far have utilized h a r m o n i c oscillator (h.o.) wave functions for the ~b,(r,-~x) in (1) or (1') above. F o r an oscillator potential with a singlenucleon p a r a m e t e r b, the c m. m o t i o n of a nucleus factors out of a completely antls y m m e m z e d shell-model wave f u n c t m n as a G a u s s l a n 7), ~, m. = ~b'"' exp

[-a(xz-mz)2/b2],

(6)

provided all shells are closed, except at m o s t one. If the tv, o nuclea have the same b, as ~s the case for identical nuclei, this leads to

G(r- m) =

(2u/re) ~ exp [ -

u(r- m)2],

(7)

where

A1A2

1 (_A 8bz

forAl=A2=lA)

u = 2 ( A a + A 2 ) bZ

Even in Investigations of the G C M where no explicit use of b o. wave functions was made, the expression (7) for G had to be assumed for any progress to be achLeved This is actually a very plausible assumption since the c.m. m o t i o n is expected to factor out. Eqs. (6) and (7) reflect the uncertainty in the position of the ions, and they are the price we have to p a y for trying to localize the c.m. We note that as the nucleus

160

H. R JAQAMAN AND A Z. MEKJIAN

becomes heawer ~t wdl oscillate less about ~ts mean c.m. and so Jt becomes more approprmte to assume that the c.m fluctuation factors out as a Gaussian that gets sharper as A increases and improvements due to G C M become less important 9) The fact that G ( r - a t ) depends on r - a t and as peaked at r - a t = 0 has been utflazed by Yukawa s) to obtain another form for the G H W equatmn. Yukawa was able to Isolate the klnetac energy of relative motmn operator m the equation, which then becomes [see Wong z)]

I

~-

where

V~- E

lJ

l(at, e ' ) f ( K ) d e ' +

v(,,

=

J

I/(at, at')f(at')dat' = 0,

(8a)

at)l Vl l (x, at')),

(8b)

EJs the energy of relative motmn, and Vt2 _- X',~l z..j~2 v(r,-rj) wath v the nucleon-nucleon force Thls alternative form for the G H W equatmn wall be more useful for the present work

3. Differential equation for g(at) Using the expression (7) for G we can write the kernels (3b) and (8b) as (v,e use K to denote L H or V) K(at, at') = f exp [ -

u(r- ~)2]k(r) exp [ - u(r- ~')2]dr,

(9)

where k(r), apart from a multlpllcatlve constant factor, is the resonating group method kernel, and ~t does not depend on either at or at'. Thas as to be compared with K(at, at) = f exp [ - 2u (r - at)z] k(r)dr

(10)

Noting that (,.-at')'- = ( r - a t + a t - a t ' )

z = ( r - a t ) z + ( a t - = ' ) Z + Z ( v - a t ) . (at-at'),

we can write K(at, at') = exp

[-.(at-at')z]fexp[-2u(r-at) z] exp [ - 2 u ( r - a t ) .

(at-~')]k(r)dr

01)

Because of the presence of exp [ - u(at - at,)z ] and exp [ - 2u(r- at)2], the contnbutton of exp [ - 2 u ( r - a t ) . (at-at')] is mainly m the region ( r - ~ ) " (at-~') ~ 0 so that we can write (up to second order) exp [ - 2 u ( v - a t ) . (at-at'J] ~ 1 - 2 u ( r - a t ) . (at-at')+Zua[(r-at) • (at_at,)]2.

NUCLEUS-NUCLEUS POTENTIAL

161

Tiffs gives

1 [v~ K(~, ~)]. v~+ 1 r(~, ~)v~

_

1 ]-V2K(a,~t)]+ 1_~ X + 16u 32u 2,,J=x,y,:

[o,a~,Sc~j

K(*t, 0t

g(a),

(12)

where differential operators contained withm square brackets operate only on the functions inside those brackets. Substituting (12) back into the GHW eq (Sa)we get

j/--h2 V_E)(Z ¼1(,,,)+ __ V,+-/(,,,)V2+I [V,/(ot,,)], 1 ~I 2/2

4u

1 IV21(,,,)] )

8u

+ (~v(',')+ l[v~ v(" ~)] "v~+ ~1

V(~,a)V~ +

1

[V2V(ot,,)])}gOt)=O,

where we have neglected the last term in (12) above Such a term would introduce an anisotropy into the effective mass. The above equation can be reduced to an equation in Ig (arguments are dropped from now on):

J-h---~v~-e }{ i +

V2

2,u

1 [~I]}

14u +

{~ +~ul [ v _~] -v ....1 [VVl [Vl]

Ig = -

28u

7u

1 V V2_ ,

14u I

I

1 V [V2I ] I V [_~]2 1 [V~V]} +--+ Ig. 14u I I 7u I 28u

I

(13)

This can be put m the form V2-q

~k2/2

14u

2/2

-

--

I

E

-I-

4/2

I

VZ+V, rr+[VU] • V - E

19 = O,

(14) where

vo.=-v+ I

1__ [ v 2 (V + h 2 [ - 7 ] ) 1 14u

4# U-

1 V

7u I

1 [V2V] + E [vZI] , I 28u I

28u

+

h 2 IV2/1

28#u

I

It is observed that (14) is a fourth-order differential equation or, in other words, it includes a velocity-dependent effective mass in addition to its radial dependence On the other hand, the ordinary Schr6dlnger equation is of second order, and if we want a description of the scattering by a potential to be acceptable, we should try to find a second-order differential equation - - - V2 + V I - E 2/2

Ig = 0 ,

(15)

162

H R. JAQAMAN AND A Z MEKJIAN

whose solutmns satisfy (14). T h e p o t e n t m l V 1 is t o be f o u n d b y substituting (15) in (14) a n d requiring self-consistency [1.e., we s h o u l d get (15) back]. T h e choice VI = (V/I)+(h2/4#)([V2I][I) is f o u n d t o satisfy the greater p a r t o f this c o n & t I o n : it exactly cancels o u t the velocity a n d r a d i a l d e p e n d e n c e o f the effective m a s s a n d the velocity d e p e n d e n t p o t e n t m l U, a n d w o u l d a p p r o x i m a t e l y satisfy the third c o n d l t m n t h a t arises" V 1 = Vef f - [V2V1]/14/.t O u r V 1 satisfies instead

14u

=VI+I

2--8-u

4## [V2I]

28u /

Therefore we can conclude t h a t (15) exists, at least to a g o o d a p p r o x i m a t i o n , ~f 28-u

V2I -

28u

<<

V + - - V2I 4/2

.

(16)

N u m e r i c a l calculations were p e r f o r m e d to check the extent to wtuch eq. (16) is satisfied for the case o f e - e scattering using the V o l k o v force no. 2 [ref. 25)] T a b l e 1 shows the values o f d such t h a t the left-hand side o f (16) is < 10 ~ a n d < 2 0 ~ o f the TABLE l The values of d as explained m the text E (MeV)

0

50

I00

150

200

250

e-~

10 ~ 20 ~

29 2 75

2 85 2.55

2.65 2.60

44 27

4.75 4 15

4.95 45

2 nuclei of mass 160

5~ 10 ~ 20 ~o

2.6 17 12

44 1.2 10

4 55 4.45 0 85

46 4.55 4.45

4 65 46 45

4.7 46 4 55

The results for mass 100 are true for separations r ~ d except r ~ 4 2 fm where the right-hand sxde of eq. (16) vanishes However, tins Js dxrectly related to tile use of the ~-ct analytic expressxon and not relevant to the two A = 100 nuclei. For both cases only separations _--<5 fm were investigated and there are always subintervals m r < d where the 5 ~o, 10 ~o and/or 20 ~ reqmrements are satisfied also

r i g h t - h a n d side for s e p a r a t i o n distances > d The left-hand side IS n o t less t h a n 5 ~o o f the r i g h t - h a n d side for a l m o s t all s e p a r a t i o n distances. I t is thus o b v i o u s t h a t (16) is n o t well satisfied for the a - e system especially since cc-a scattering is sensitive t o the inner region o f the potential. A l s o s h o w n in table I are similar results o b t a i n e d f r o m the same analytic expression used for the a - e case b u t with larger values o f # a n d u a p p r o p r i a t e for a system o f two nuclei o f mass 100. It is a p p a r e n t t h a t u p to an energy ,.~ 100 M e V eq (16) is m u c h better satisfied for this case, especially in the tail region o f the p o t e n t i a l ; this region is a l l - i m p o r t a n t in h e a v y i o n scattering. This agrees with w h a t is expected f r o m e x a m i n i n g eq. (16) itself where, for larger/~ a n d u, all the terms w o u l d b e c o m e smaller c o m p a r e d to V.

NUCLEUS-NUCLEUS POTENTIAL

163

We can now make the following further observations: (a) The above equations [(13), (14) and (15)] are for I(c¢, ~)g(~) and not for y(~), but since I(~, ~) --, 1 for large 0c (the only region where g(~) has to be known) we can regard I(~, x)#(~) as our wave function This ambiguity m the wave function at short distances is a reflection of the fact that the kinetic energy operator IS not unique at short distances (with a complementary ambiguity in the effectwe potential) and as a consequence of antlsymmetrizatlon [see ref. 2) p. 337]. It is also intimately connected with the removal of velocaty-dependent potentials 13) so that I(~, c¢) here plays a role similar to that of F m ref 13) Tlus can be seen from (15) which, when changed to an equation for 9(~) alone, would get a new term added to the potential plus a velocity-dependent term (b) The potential V 1 can be looked upon as equivalent to the real part of an optical potential and can be compared with phenomenologlcal optical potentials It consists of two parts V 1 = (V/l)+(h2/41.t)(V2I/I), the first part is the potential that one would get m a Heatler-London approximation 14) and the second part has a form that suggests it is a kinetic energy contribution to the potential for relative motion between the two nuclei. There has been some discussion 15) by Zlnt and Mosel of such a contrabutaon, but whereas these authors predict a repulsive contribution (based on the Born-Oppenheimer approximation), the term (h2/4~t)(V2I)/I is not repulsave everywhere In fact at is attractive for part of the tad region, although it :s neghgible there (c) There have been some objections l o) against the interpretation of V a (or s,irnlar terms) as being equivalent to the real part of an opt,cal potentml These objections depend mainly on model calculations that show a strong varmtlon of ~lCfr m the anteractlon region 11). We have a few observations about this point" 0) The calculations in ref. 11) do not include the velocity dependence of the effectave mass This dependence [as those authors observe 11)] as important for heavy ion ~cattering and neglecting at would amount in eq (14) above to omitting the V2V 2 term which in turn would make it amposslble to get to eq (15) where the effective mass IS a constant (11) Those calculations ~ere done using the cranking formula xxhose validity ~s questionable for heavy aon reactions. Thls formula applies to adiabatic processes where at as safe to assume that the velocity of the collective motion (m the present case this would be the relative motion between the two heavy Ions) is small compared with the Fermi motion 12) Th~s is why the cranking formula has a perturbatlon-theoretacal character (m) There is no reason why effectave masses obtained in different models or by dafferent processes should be the same, since these masses have no physical significance by themselves, and are not necessardy defined In the same way in all models. The effectwe masses obtained in thas work do not show any of the strong r-dependence or the order of magnitude devmt~on from the reduced mass # that was found m r e f . 11). Even if we Include the h~gher-order terms that were omitted an getting to (12) and (13) no major changes in the above conclusions are expected, since the important terms are all in (13) already However, ~t

t64

H R JAQAMAN AND A Z MEKJIAN

could stdl be the case that the objections raised m r e f . ~o) are s:,Jl valid where a potential description is not possible [Le, where (16) is not satisfied].

4. The real part of the optical model potential 4.1 GENERAL DISCUSSION In sect. 3 It was remarked that V I can be regarded as the real part of an optical potential for the elastic scattering of heavy ions and that the term VII m VI corresponds to the Heitler-London approximation. This approximation amounts te putting A~ nucleons m one well to form one nucleus and A2 nucleons m another well, at a chstance at from the first, and then calculate the lnteractmn between the two nuclei This is to be contrasted with the two-center shell model (which is eqmvalent to the molecular-orbital method) m which each nucleon is s:~pposed to be m the field of both wells Denoting the nucleons m nucleus 1 by the numbers 1 -~ A 1, and m nucleus 2 by A 1 + 1 -* A1 +A2, we define

s,,

2,, 2s = 1, 2;

= j ¢*(r, - at~,)¢~(,2 - at~,)v(at +,2 - r,)~k(,i - atO¢,(r2 - at0d'i dr2, g < AI,

J > AI,

at = atz,-at~s,

and define D . , kt as the determinant formed from [Sp~] by omitting the ith and j t h rows and the kth and lth columns, mult:plied by the p a n t y of the transformation that brings the ith a n d j t h rows to the posmons of the first and second rows, and the kth and lth columns to the posmons of the first and second columns. The two vertical hnes in IS,~l denote the determinant of the matrix S w. The notatmn here ~s s~mllar to that of Slater 14) but not identical In what follows t ____AI, A1 < j <_ A I + A 2 m all expressmns, With these definmons we have i4)

v(at, at) = <~"~(~, at)lVx21~(x, at)> = ,j~, I(at, at) ( ~(a)(x, ae)l~(x, at))

(ij]vlkl)D,j,u ISpqI

Some terms vanish, namely those corresponding to exchanges between nucleons belonging to the same nucleus, and we get

V _ I

1 ~ {[(iJlvlij)-(iJIvlji)]D.,.+ ~ (ijlvlkl)D.,kt}, IS,~l ,,J

(17)

k~

for at least one of k, l ~ i, j. In the important tall region, where there IS no substantial overlap between the two ions, the above expressmn s~mphfies considerably since all the exchange terms be-

H R. JAQAMAN AND A Z MEKJIAN come negligible, Spq ~ ~Spq,ISp~l ~ 1, V, --, V ~ E I ,j

D,j, ,~ ~

165

1 and

<,Jlvl,j> = ~P,(rl)p2(r2)v(~+r2-rl)dr, d

dr2,

(18)

i e , we end up with a folding model for the potential that has been previously introduced, more or less, phenomenologically t6,tT) It is also worthwhile to study (17) assulmng it apphes for all separation distances between two identical ions. As ct ~ 0, ISpql -* 0 and, if we use as we should, a nucleon-nucleon force v that allows for nuclear saturation [a density-dependent force ,7, is) for instance], this will lead to a steeply repulsive core which prevents the two ions from completely overlapping, as expected by the Pauh principle. This repulsion can be related to non-physical states excluded by the Pauh pnnclple, the so-called null states 2.s) For the sake of comparison, we define also an adiabatic hmlt for V/I, considering (17) as discussed above to be some sort of a non-adiabatic or sudden limit for the potential. Actually, the G C M , on which (17) is based, is intimately connected to a sudden-approximation picture of the interaction between the two nuclei: freeze the internal wave function of each nucleus, calculate the kernels and then allow the nucleons to interact. In the adiabatic limit we assume that the two nuclei are at rest with respect to each other at a certain distance, and allow their internal wave functions to readjust as required by the exclusion principle, the wave functions of the nucleons in one nucleus can be expanded in terms of only the unoccupied eJgenstates of the H a m d t o m a n of the other nucleus. Then S,j = 6,~, I = 1, and

V,(,) = ~ [- ] *l

=

f p,(r,)pz(r2)v(=+ r.,-r,)dr, dr 2 -

~'



(19)

tJ

For ~t = 0 and a zero-range force the exchange contnbutlon is ¼ the direct contrlbuuon to VI [since < q l v l J * > = f o r ~ = 0 and i,j having same spin and lso~pm] and we get

1,'1(0) = ¼ | p,(r,)p2(r2)v(r 2 - r,)dr,

dr 2 .

(20)

d

As ~ increases the exchange integrals become less and less important until they become negligible in the tall region so that the folding expression (18) emerges again Two points are worth noting about (19), 0) it does not include the effect of the null states and this has to be kept in mind when using the expression (19) (see subsect 4 2), and 0i) pt and P2 are not free-nucleus densities, except in the tall region of the potential where there is very little overlap, but are the Instantaneous densities that include the deformation that the nuclei induce in each other. These are obviously difficult to calculate and are similar to the two-center shell-model densities As a

166

H R J A Q A M A N A N D A Z. M E K J I A N

first approxamatton, however, they can be taken as the free-nucleus densatles ("frozen" density approxlmat~on) Unhke (17), (19) does not have a steeply repulsave core, af any. For v, the nucleon-nucleon force, ~t ~s customary to use one of the effectave rateractions, such as the Skyrme force 19), used an nuclear structure calculataons. Howevcr, for scattering calculations we have to add to thas force a non-hermatean term that accounts for compound nucleus formation 2o). Such a term would mainly contrabute to the imaginary part of the potentml, is difficult to calculate so that a paramcrazed form is used instead, and would be neglected here as far as the real part of the potentml is concerned. Furthermore, all the precedang dlscussaon has been earned out m a one-channel pacture, whereas in practice a number of channels are usually open These can be taken into account by a coupled-channels calculation 2a) which adds another term to the elastac channel effective operator, its real part will also be considered neghgable The neglect of all the above terms as justified in the surface region by the success of such phenomenological potentials as the foldang model. The above dtscusslon throws some hght on possible sources of an energy dependence m V1. A m o n g these we identify: (1) The energy dependence of the effectave mteractaon itself in addition to the energy dependent terms added to at m order to account for compound nucleus formation and open channels (including both the non-locahty of these terms and the increase an the number of open channels wath energy) (li) The Pauh principle whose role changes with energy. (lIl) The energy dependence that is expllcltly found in the expressaon for V~ff m eq. (14), which would be felt ff eq. (16) is not well satisfied Thus at as difficult to pre&ct the energy dependence of the potentml unless at is the case that one of the above sources dominates all the others. 4 2. C A L C U L A T I O N S W I T H A D E N S I T Y - D E P E N D E N T I N T E R A C T I O N

In a prevaous work a 7), the present authors used a densaty-dependent zero-range nuclear force hnear in the local density. v(ra - r2) = - ~o 6(rl - r2) + flP(½(rl + r2))6(rl

- r2).

(21)

Such a force can be regarded as a very sampllfied form of the Skyrme mteractaon (which atself can be used if desired), and ~s chosen for the ease at provides in carrying out calculations and obtaining analytic results and ~ts abdtty to reproduce important nuclear effects, especaally saturation. A more sophastacated force is clearly needed for accurate calculations. It was shown a 7) that by choosing ~o = 956 MeV • f m 3 and flo = 2175 M e V . fm 6, (21) gaves a reasonable value for the tall of the potential obtained m a foldang procedure (eq. 0 8 ) ) . This does not provide a sensltave test of the effect of the density-dependent part of (21) because of the httle overlap between the densities of the two nuclei m th~s case. A better test would revolve the inner region of the potentaal where there is an appreciable overlap between the nuclez Such a

NUCLEUS-NUCLEUS POTENTIAL

167

situation is prowded by cluster-model energy levels m nucleL For th~s case ~t Is reasonable to use the admbatlc hmlt expression (19) Vary and Dover ~6) used this expression for the calculation of energy levels of ~clusters on ~60 in Z°Ne with a density-independent delta interaction, but they had to introduce a considerable renormahzatlon of the strength of the lnteractaon (as compared to the one used for scattering calculations) m order to obtain a qmte successful reproduction of the bound and resonant energy levels m ZONe. The use of density-dependent forces explains the major part of the renormahzat~on needed Substituting (21) in (19) we get VI(~) = i { - S o f Pl ( rx ) P2 ( r l - a t ) drl+-~flo f P~( r l ) P2 ( r l - - g )Pl°¢darl} ,

( 22 )

where P~oc as the local density at the s~te of interaction, the factor ¼ s~mulates the effect of the Pauh principle, and the ~2 factor ~s due to the rearrangement effects 22) that result from using a density-dependent force. Both factors are exact at the origin and are a fairly good approximation m the region near the o n g m that determines the depth of the potential. They go to 1 as the separation increases. The first problem that arises is what to use for p~o¢. A first guess might be the sum of the densities Px and P2 at the site of the mteractton, however, thas ~s not the case. Even where the sudden approximation is apphcable, Paoc = Pl + p z + m t e r f e r e n c e terms, since we should really add the wave functions and not the dens~tles In addition to the interference terms, Pi and Pz m~ght be d~storted from the free state dens~taes On the other hand, we have to keep m mind that the two clusters (a and x60) are part of the nucleus Z°Ne which, as a first approximation, can be thought of as a uniform sphere of density Po- Neglecting edge effects, (22) reduces to 3

where, now, cq = ~ o - ~ z f l o P o . Putting Po ~ 0.16 fm -3, cto = 956 M e V - fm 3, flo = 2175 MeV • fm 6, we get. cq ~ 434 MeV • f m 3 to be compared with the value used by Vary and Dover x6) ,

~1 -

4 2rrh z

3

M

f-

4 2n(197) 2 3

940

1.237 = 4 2 7 M e V

fm 3,

while the value they used for scattering calculations is -,~s o So whde the very close agreement between ~ and ~1' could be accidental, xve can at least conclude that we are able to account for the magmtude of the normahzmg "strength f a c t o r " f that they determine phenomenologlcally. The null states are taken into consideration by making sure that the calculated wave functions have the correct number of nodes, e.g, the 0 + state m the ground state band should have four nodes, etc. On the other hand, if we use (17) we do not do this, the afore-mentioned 0 + state will have no nodes

168

H. R J A Q A M A N A N D A. Z. M E K J I A N

The repulswe core suppresses the tuner oscillations of the wave function [see ref. 23) for a smular discussion about 160-160 scattering]. We now consider a classical-geometrical model in whtch the two overlapping nuclei remain segments of uniformly dense spheres whose radn are functions of the distance between the two centers of the nuclei under the assumptmn that the enclosed volume is a constant (lncompressiblhty) This g~ves the equatmn (see fig. 1): 3

3

1 3

R 1+ R z - ~ r

+¼r(R21+ R 22) +

3 8rr (R2-R2)2 = 2(R°al +R°a2)"

Exammatmn of th~s equation m&cates that R2 + Rt + {R~I +Ro*2}~ as r -+ 0, i.e., the radii of the spheres become larger. For two identical spheres R

--~ Ro

2~ -

1

r

---+ 4 R0

1

1

r2

2 ~ 16 R 2

+

1

1

rs

+ ....

96 2 ~'Ro3

For Rol ~> Ro2, the smaller sphere undergoes a more drastic change than the larger one.

Fig. 1. Continuous hne shows the original size o f the nuclei, and the dotted hne the sizes w~th the assumptmn o f no compression.

Motivated by the above picture, we can get a faarly good approximation for the local density when the two nuclei have a large overlap by giving them larger effective radii (and a smaller central density to conserve the total mass), and then putting Pl.c = Pl +P2. It is interesting that exact calculations 24) ofpl.o that include the interference effects tend to agree quahtatwely with this picture, the nuclei appearing to have larger radi~, in the static hrmt that apphes when the relatwe motmn between the two nuclei can be neglected. Harmonic oscillator densities are used for 160 and ct:

p,60(r)=4(~)~(l+2vlr2)e -~1:, (24)

with v I = 0.314 and v2 = 0.538 fm -a for the free nuclei. To get larger radii, h o b ever, we have to use smaller values for vl and v2, which should be functions of the

NUCLEUS-NUCLEUS

POTENTIAL

169

MeV 0 -I -p

-5

2 + __...__s

I

I~"

~ x _.....~ x

/

-4-

]"

0+--

t

-5

a)

b)

c)

d)

e}

Fig 2. Energy levels for the Z°Ne g r o u n d - s t a t e band. (a) E x p e r i m e n t a l results. (b) R e s u l t s u s i n g eq. (23) w R h m, = 417 M e V - f m 3. (c) R e s u l t s u s i n g eqs. (22) a n d (24) w R h P~oo = pl-kpz a n d 1,1 ----vz = 0.314. (d) S a m e as (c) b u t wRh vl = 0.314 a n d ~z ---- 0.3. (e) S a m e as (e) b u t wlthv~ = ~'z = 0.3. N o t e t h a t it is posmbl¢ to fit each level by a s h g h t r e a d j u s t m e n t o f ~,~ a n d ~z a n d t h a t this gives t h e correct d e p t h for t h e potential. Energies are m e a s u r e d with respect to the c<-t60 threshold, a n d only b o u n d levels are conmdcred.

0 -201

-40

o~-t60 potenhal

-60 (1.) v

-80 -I00 -120

J

/// / / /

-140

/ /

-160

t

/

/ i

I

I

I

)

2

+

,~

5

+

t

I

[

I

e 7 r (fro)

t

+

9

to

+l

~2

Fig. 3 T h e real part o f t h e ~-160 potential m t h e adiabatic h m l t . 0 ) c o n t i n u o u s curve corr e s p o n d s to (c) m fig. 2, a n d 01) d a s h e d curve c o r r e s p o n d s to (b) m fig 2.

170

H. R J A Q A M A N A N D A Z M E K J I A N A,IO9

160-160 potenhal

~

\

\

'~

l /

,oo

o

>

I\

,

,

/,/

/,/

- oo

/

(~)/" I

/ (c)~lo) / /

,'t

//,

// /

-500

I

/lb~ J

I

l

2

~'1

5

,

///{//

~_10~ / . / /

-I00

- 200 >

,/

/lbt~lo I T

)

I

4

5

G

-(

'

[

I

I

l

8

9

I0

II

12

r (fm) Fig. 4 T h e 16Oo160 potentml. (a), (b), a n d (d) are calculated m the admbatac h m l t ; (a) is v a h d near the origin, (b) a n d (d) are v a h d m the t a d r e g m n [the actual p o t e n t m l curve m t h e m t e r m e d m t e r e g m n lying between (a) a n d (d)] (c) c o r r e s p o n d s to (eq (17)), 'the s u d d e n app r o x i m a t i o n ' (b), (c), a n d (d) are also magnified ten times for r > 5 5 f m to s h o w t h e t a d r e g m n m o r e clearly. (e) is the c o r r e c t m n that h a s to be a d d e d to (c) to give Vz (eq (15)) T h e ~ 6 0 + x 6 0 elastic scattering is sensitive to the p o t e n t m l for r > 6 5 f m only

V/I

(~2/4#)(VzI/I)

separation distance. In practice, t h i s is not easy to do and constant values are chosen so that the resulting potential is probably not acceptable for the surface region Results for cluster-state calculations are shown in fig. 2. Fig. 3 gives the correspondmg ~-t 60 potentials. The t 60-a 60 potential is also calculated and shown in fig. 4 where curve (a) IS the potential obtained when eq (23) is used with ~t = 420 M e V . fm 3 (this potential is good near the origin and gives the depth of the potential), (b) is the potential obtained from eq (18) using (21) with ~o = 956 MeV • fm 3 and neglecting the density-dependent part since the resulting potential is good only in the tall region where this part is neghglble Curve (c) is the t 60-~ 60 potential according to eq. (17); It is taken from the work of Yukawa 8) who uses a Volkov i s ) effective force which has a finite range, and is shown here for comparison The longer range of (c) In the far tall region is due to the use of a finite-range force in the calculation (as opposed to the zero-range force used for (b)). The difference between (b) and (c) for r < 7 fm ~s due to the neglect of the density-dependent part of the delta interaction and the omlssmn of exchange effects. This is obvious from curve (d) which is the same as (b) except that the density-dependent part of the force is Included. Also shown in fig 4, curve (e), is the kinetic energy term (h2/4#)(V2I/I) that has to be added to VII (curve

NUCLEUS-NUCLEUS POTENTIAL

171

(c)) to form the optical potential Vx (see eq (15) and the discussion following it). It ~s apparent that this term modifies the potentml and the final result ~s much shallower than without this term [cf. ref. 15)]. In the ~-160 calculations, the Coulomb potential used was that due to the interaction between two spheres 26) which is probably better than the sphere-point charge potential usually used However, the energy level splitting is not sensitive to the Coulomb potential, and choosing different forms for the Coulomb potential merely shifts their position, which can be compensated for by a slight readjustment of the depth of the nuclear potential

5. Conclusion We have shown that the G C M for heavy ion reactions is equivalent, to a good approximation, to solving a Schrodlnger equation with an effective potential whose real part has mainly a sample expression and includes a kinetic energy contribution besides the usual term that results from the interaction between the nucleons belonging to the different nuclei. This lmphes that a potential description of heavy 1on reactions is not without a physical bas,s, and It explains the fact that phenomenolog~cal optical potentials have been very successful. The G C M has been a valuable tool, and thxs would become more apparent when calculations for heavier nuclei are performed. The simple density-dependent delta force we used has proved to be quite successful m reproducing many results.In spite of all the simplifications introduced the potential obtained stdl retains most of the properties of interest, and its simplicity gives a better feel for the interaction between nuclei, at least as a first step towards a comprehensive and more accurate unified microscopic description The authors would like to thank Profs L Zamlck and S Yoshlda for some interesting and helpful dlscussmns

References

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172

H. R JAQAMAN AND A. Z. MEKJIAN

11) P. Llchtner, D. Drechsel, J. Maruhn and W. Greiner, Phys. Rev. Lett. 28 (1972) 829 12) L. Zamick, invited talk, Tucson Conf. on effective interactions and operators, University of Arizona, June 2, 1975 (preprint); Y. M. Engel et aL, Umversity of Oxford, Department of Theoretical Physics, preprmt 13) F. G. Percy, in Nuclear spectroscopy and reactmns, ed. J. Cerny (Academic Press, New York, 1974) p. 145 14) W. Heltler and F. London, Z Phys 44 (1927) 455; H. Margenau, Phys. Rev. 59 (1941) 37; J. C. Slater, Quantum theory of molecules and sohds (McGraw-Hall, New York, 1963) voL 1 15) P. G. Zmt and U. Mosel, Phys. Lett. B56 (1975) 424 16) J. P Vary and C. B. Dover, Phys. Rev. Lett. 31 (1973) 1510, G. W Greenlees, G. J Pyle and Y. C. Tang, Phys. Rev. 171 (1968) 1115, D° M Brink and N. Rowley, Nucl. Phys 3.219 (1974) 79 17) H Jaqaman and A. Z. Mekjlan, Phys. Lett. 56B (1975) 237 18) B. Smha, Phys Rev. Lett. 33 (1974) 600 19) T. H. R. Skyrme, Nucl. Phys 9 (1959) 615 20) N. Austern, Direct nuclear reaction theories (Wdey, New York, 1970) p 63 21) H. Feshbach, Ann. of Phys. 5 (1958) 357; 19 (1962) 287 22) R. Sharp, Ph. D. Thesls, Rutgers Umverslty (unpubhshed) 23) A. Tohsaki, F. Tanabe and R. Tamagakl, Prog. Theor. Phys. 53 (1975) 1022 24) T. Fhessbach, Z. Phys. 238 (1970) 329,242 (1971) 287 25) A. B. Volkov, Nucl. Phys 74 (1965) 33 26) T W Donnelly, J. Dubach and J D. Walecka, Nucl. Phys. A232 (1974) 355